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Two-photon-resonant third-order nonlinear susceptibilities in parabolic quantum well wires Kang-Xian Guo*, Chuan-Yu Chen Department of Physics, Guangzhou Normal University, Guangzhou 510400, People’s Republic of China Received 5 January 1998; received in revised form 4 May 1998; accepted 23 July 1998

Abstract Two-photon-resonant third-order nonlinear susceptibilities in parabolic quantum well wires due to resonant intersubband transitions are analyzed with a compact density-matrix approach. Analytic expressions for the two-photonresonant third-order nonlinear susceptibility are derived. The numerical results are presented for a typical GaAs parabolic quantum well wire. 1999 Elsevier Science B.V. All rights reserved. PACS: 78.20.Bh; 71.38.#i Keywords: Two-photon resonance; Parabolic quantum well wire

1. Introduction With the recent advances in the epitaxial techniques such as molecular beam epitaxy (MBE), metal-organic chemical vapour deposition (MOCVD), and liquid-phase epitaxy (LPE), it has been possible to confine electrons in extremely thin semiconducting wires, namely, quantum-well wires, with submicron dimensions. Due to their small size these quantum-well wire structures present some physical properties that are quite different from those of the semiconductor constituents such as optical and electronic transport characteristics. In these quasi-one-dimensional structures the electron motion along the length of the wire is quasi-free, but it is quantized in the two dimensions perpendicular to the wire. A great deal of theoretical and

* Corresponding author.

experimental interest has been devoted to the study of the nonlinear optical properties of these onedimensional semiconductor systems [1—7]. It is expected that the optical nonlinearities are more sensitive to non-square quantum well shapes than to square well shapes [3]. In comparison to square quantum wells, parabolic quantum wells present properties such as a nearly uniformly spaced density of states for the electrons and holes. Quantum confinement of carriers in a semiconductor parabolic quantum well leads to the formation of discrete energy levels and a drastic change of optical susceptibilities [4]. Two-photon-resonant third-order nonlinearities have widespread practical applications; this is because two-photon resonance can significantly enhance the desired nonlinearity, whilst at the same time competing absorption processes can be minimised by avoiding coincidences between the optical frequencies and single-photon resonances. To our

0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 4 7 6 - 1

K.-X. Guo, C.-Y. Chen / Physica B 262 (1999) 74—77

knowledge, the two-photon-resonant third-order nonlinearities in parabolic quantum well wires have not been studied. In this paper, we further investigate the two-photon-resonant third-order nonlinear susceptibilities in parabolic quantum well wires. In Section 2, the formula for the two-photonresonant third-order nonlinear susceptibility is given. In Section 3, the numerical results for a GaAs parabolic quantum well wire are presented.

2. Theory Let us consider a parabolic quantum well wire extending along the z direction with a parabolic cross section. The electron is confined to move within the x!y plane, but free to move in the z direction. The parabolic well exists in the GaAs region of the superlattice. The Hamiltonian for this system can be expressed as

j j j H "! * # # #»(x, y), 2m jx jy jz (1) »(x, y)" m*u (x#y), where we assume that u "u "u , m* is the V W effective mass of the electron in GaAs; m*ux and m*uy are, respectively, the parabolic confining potential in the quantum well in the x and y directions. Since the electron is almost free in the z direction, we utilize a plane-wave representation for its motion in the z direction, i.e., we take (z) & e IX. The eigenfunctions W(x, y, z) and eigenenergies E are solutions of the Schro¨dinger equation H W(x, y, z)"EW(x, y, z), where,

(2)

W(x, y, z)" (x) (y) (z), L LY a

(x)" e\?VH (ax), L L p2Ln!

(3)

(7)

Here, a"(m*u / ); H (ax) H (ay) are both Her L LY mitian functions; º (z) is the periodic part of the Bloch function in the conduction band at k"0. If the system is excited by an internal electromagnetic field E (t)"E cos ut, the electronic polariza tion of the parabolic quantum well wire can be expressed as P (t)"P (t)#P (t)#P (t)#2 "e (sE(t)#sE(t)#sE(t)#2),

(8)

where s and s denote, respectively, the linear susceptibility and second-order nonlinear susceptibility; s is the third-order nonlinear susceptibility discussed in this paper. Here we only consider the two-photon-resonant third-order nonlinear susceptibility s(!u;!u, u, u). The Hamiltonian of this system excited by the internal electromagnetic field E(t) can be split into two parts as H"H #»(t),

(9)

where H represents the unperturbed Hamiltonian, and »(t) represents the interaction Hamiltonian which is given by »(t)"!exE(t).

(10)

The evolution of the density matrix o obeys the LK following equation: jo 1 LK" [H, o] ! C (o ! o) , LK LK LK jt i

(11)

o "o#o#o#2 LK LK LK LK

(12)

with (4)

a

(y)" e\?WH (ay), LY LY p2LYn! (n"0, 1, 2,2),

E" u (n#n#1)# *k. 2m

(6)

which is solved using the iterative method:

(n"0, 1, 2,2),

(z)"º (z) e IX A

75

(5)

joH> 1 LK " +[H , oH>] ! i C oH>, LK LK LK jt i

1 ! [ex,oH] E(t). LK i

(13)

76

K.-X. Guo, C.-Y. Chen / Physica B 262 (1999) 74—77

The nonlinear susceptibility of the jth order sH is given by 1 sH" ¹r(oHex), (14) e p EH(t) where p is the density of electrons. After tedious and complicated calculations [8], we find the twophoton-resonant third-order nonlinear susceptibility s(!u;!u, u, u):

10 m\, the lattice constant a"5.654 As ; for simplicity, we choose C\ to be 0.2 ps. LK Fig. 1 shows the two-photon-resonant third-order nonlinear susceptibility "s(!u;!u, u, u)" versus the confinement length ¸ (¸"( /m*u ) of the parabolic quantum well wire for the photon energy u"23.40 eV. From Fig. 1, we can see that the two-photon-resonant third-order nonlinear

5p e k s(!u;!u, u, u)" k k k k 3e [(u !u)!ic ][(u !2u)!ic ][(u !u)!ic ] k # [(u !u)!ic ][(u !2u)!ic ][(u #u)#ic ] k # [(u !u)!ic ][(u #2u)#ic ][(u !u)!ic ] k # [(u !u)!ic ][(u #2u)#ic ][(u #u)#ic ] k # [(u #u)#ic ][(u !2u)!ic ][(u !u)!ic ] k # [(u #u)#ic ][(u !2u)!ic ][(u #u)#ic ] k # [(u #u)#ic ][(u #2u)#ic ][(u !u)!ic ] k # , (15) [(u #u)#ic ][(u #2u)#ic ][(u #u)#ic ]

where C is the relaxation rate, u is Bohr’s LK LK frequency, k ""1 (x)"x" (x)2", (n, m"0, 1, 2, 3; nOm), LK L K (16) k ""1 (y)"y" (y)2", (n"0, 1, 2, 3). LL L L

(17)

3. Results and discussions Now let us choose a GaAs parabolic quantum well wire as an example to present the numerical results. In the calculations we have used the following parameters of GaAs: e "12.83, p "5;

susceptibility "s(!u;!u, u, u)" has a very strong resonant peak at ¸"53 nm. As the confinement length ¸ shifts to the left or right of the dot ¸"53 nm, the resonant peak rapidly weakens till it disappears. This feature indicates that the twophoton-resonance can significantly enhance the third-order nonlinear susceptibilities; particularly, when the photon frequency u is close to the confinement frequency u in parabolic potential, "s(!u;!u, u, u)" is very strong; otherwise, it is very weak. Fig. 2 shows the dependence of the two-photonresonant third-order nonlinear susceptibility "s(!u;!u, u, u)" on the normalized photon

K.-X. Guo, C.-Y. Chen / Physica B 262 (1999) 74—77

77

Fig. 1. Two-photon-resonant third-order nonlinear susceptibility "s(!u;!u, u, u)" versus the confinement length ¸ (¸"( /m*u ) of the parabolic quantum well wire with the photon energy u"23.40 eV.

Fig. 2. Two-photon-resonant third-order nonlinear susceptibility "s(!u;!u, u, u)" versus the normalized photon energy

u/d, for three different confinement lengths ¸ of the parabolic quantum well wire: (a) ¸"50 nm, (b) ¸"100 nm, (c) ¸"200 nm.

energy u/d, where d is the energy separation of the highest valence-band state and the lowest conduction-band state. It is plotted for three values of the confinement length ¸ of the parabolic quantum well wire: (a) ¸"50 nm, (b) ¸"100 nm, (c) ¸" 200 nm. This figure clearly shows resonance features at photon energies near 0.39 d, 0.45 d, 0.52 d, respectively, for ¸"50 nm, ¸"100 nm, and ¸" 200 nm. In particular, the smaller the confinement length ¸, the sharper the peak, and the larger the peak intensity. Another important feature is that the peak will shift to the right of the curve, and will disappear gradually, when the confinement length ¸ increases. Finally, we hope that this paper would stimulate more experimental work being helpful on study of the two-photon-resonant third-order nonlinear susceptibilities in parabolic quantum well wires.

Acknowledgements One of the authors (Dr. Kang-Xian Guo) acknowledges the support of the Natural Science Foundation of Guangdong Province under Grant No. 970285. References [1] P.M. Petroff, A.C. Gossard, R. A. Logan, W. Wiegmann, Appl. Phys. Lett. 41 (1982) 645. [2] R. Chen, D.L. Lin, B. Mendoza, Phys. Rev. B 48 (1993) 11879. [3] A.C. Gossard, R.C. Miller, W. Wiegmann, Surf. Sci. 174 (1986) 131. [4] R. Dingle, W. Wiegmann, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. [5] K.X. Guo, S.W. Gu, Solid State Commun. 87 (1993) 741. [6] K.X. Guo, Solid State Commun. 103 (1997) 255. [7] W. Wu, Phys. Rev. Lett. 61 (1988) 1119. [8] R.W. Boyd, Nonlinear Optics, Academic Press, New York, 1992.

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